Related papers: A probabilistic approach to some binomial identiti…
By applying the derivative operators to Chu-Vandermonde convolution, several general harmonic number identities are established.
We prove an interesting identity for the sum of determinants, which is a generalization of the sum of a geometric progression. The proof is quite long and a number of other identities are proved along the way. Some of the more elementary…
We derive some Fibonacci and Lucas identities which contain inverse binomial coefficients. Extension of the results to the general Horadam sequence is possible, in some cases.
We present a different combinatorial interpretations of Lucas and Gibonacci numbers. Using these interpretations we prove several new identities, and simplify the proofs of several known identities. Some open problems are discussed towards…
A generalization of the Catalan numbers is considered. New results include binomial identities, recursive relations and a close formula for the multivariate generating function. A simple expression for the Catalan determinant is derived.
This paper presents new identities expressing the terms of Fibonacci, Lucas, and generalized Fibonacci sequences with multiple indices through powers of Lucas numbers and binomial coefficients. The obtained formulas rely on the application…
In this paper, a class of combinatorial identities is proved. A method is used which is based on the following rule: counting elements of a given set in two ways and making equal the obtained results. This rule is known as "counting in two…
In this paper, we introduce a novel identity for generalized Euler polynomials, leading to further generalizations for several relations involving classical Euler numbers, Euler polynomials, Genocchi polynomials, and Genocchi numbers.
This note is dedicated to Professor Gould. The aim is to show how the identities in his book "Combinatorial Identities" can be used to obtain identities for Fibonacci and Lucas polynomials. In turn these identities allow to derive a wealth…
We present here some new identities for generalizations of Fibonacci and Lucas numbers by combinatorially interpreting these numbers in terms of numbers of certain tilings of a $1 \times m$ board. As a consequence, some new interesting…
In the paper we provide some polynomial identities for finite-dimensional algebras. A list of well known single polynomial identities is exposed and the classification of all $2$-dimensional algebras with respect to these identities is…
We derive several symmetric identities for Bernoulli and Euler polynomials which imply some known identities. Our proofs depend on the new technique developed in part I and some identities obtained in [European J. Combin. 24(2003),…
When searching for Calabi.Yau differential equations, often different formulas for the coefficients give the same differential equation. The coefficients are usually sums (simple, double or triple) of products of binomial coefficients. This…
Recently, the numbers $Y_{n}(\lambda )$ and the polynomials $Y_{n}(x,\lambda)$ have been introduced by the second author [22]. The purpose of this paper is to construct higher-order of these numbers and polynomials with their generating…
In this paper we derive some interesting identities arising from the orhtogonality of gegenbauer polynomials.
The aim of this article is to promote the use of probabilistic methods in the study of problems in mathematical general relativity. Two new and simple singularity theorems, whose features are different from the classical singularity…
We consider binomial and inverse binomial sums at infinity and rewrite them in terms of a small set of constants, such as powers of $\pi$ or $\log(2)$. In order to perform these simplifications, we view the series as specializations of…
In combinatorics, the probabilistic method is a very powerful tool to prove the existence of combinatorial objects with interesting and useful properties. Explicit constructions of objects with such properties are often very difficult, or…
Using elementary means, we prove several identities involving the M\"obius function, generalizing in the multidimensional case well-known formulas coming from convolution arguments.
We give a general multiplication-convolution identity for the multivariate and bivariate rank generating polynomial of a matroid. The bivariate rank generating polynomial is transformable to and from the Tutte polynomial by simple algebraic…